DepartmentofMathematicsandStatisticsIndianInstituteofTechnologyKanpurMSO202A/MSO202Assignment4IntroductionToComplexAnalysis

Theproblemsmarked(T)needanexplicitdiscussioninthetutorialclass.Otherproblemsareforenhancedpractice.

1. (T)Giveexamplesforthefollowing:

(a)TheradiusofconvergenceofTaylorseriesofafunctionwithcenterassomepoint a inthedomainofanalyticityD ofthefunctionislargerthanthelargestdisk z a R containedinD

(b)TwoTaylorserieswithdifferentcentersrepresentthesameanalyticfunctionintheintersectionoftheirdisksofconvergence.(c) The disk of convergence of Taylor series of a function is strictly contained in the domain ofanalyticityofafunction.

2. Evaluatethefollowingintegralsontheindicatedcurves,allofthembeingassumedtobeorientedinthecounterclockwisedirection:

(T)(a) 4

1, : 2

1C

dz C zz

(b)

3 2

4 2

2 4, : 2 4.

4C

z zdz C z

z z

3. Evaluate the following integrals on the square C, oriented in the counterclockwise direction and

havingsidesalongthelines 2 2x and y :

(T)(i) 2

cos

( 8)C

zdz

z z (T)(ii) 4

cosh

C

zdz

z .

4. UsingLiovuilleTheorem,showthatthefunctionsexp(z),sinz,cosz,sinhz,coshzarenotboundedin

thecomplexplaneC.5. ShowthateverypolynomialP(z)ofdegreenhasexactlynzerosinthecomplexplane.6. If f is an entire function and 0( ) nf z MR in z R , prove that f is a polynomial of degree

atmost 0n .

7. Let ( )f z beanalyticin z R .Provethat,for0<r<R,

2 2 2

2 20

1( ) (Re )

2 2 cos( )i iR r

f re f dR r Rr

(calledPoissonIntegralFormula).

8. (T)Evaluate 4

1dz

z ,where isthepartofclockwiseorientedellipse

2 2( 3)1

1 4

x y lyinginthe

upperhalf‐plane{ : Im 0}z z .

P.T.O.

29. Findtheorderofthezeroz=0forthefollowingfunctions:

22( ) ( 1)zi z e (T) 3 3 6( ) 6sin ( 6)ii z z z (T)(iii) sin tanz ze e 10. Findtheorderofallthezerosofthefollowingfunctions:

( ) sini z z (T) 2 3( )(1 )( 4)zii e z (T)3sin

( )z

iiiz

11. (T)Doesthereexistafunctionf(z)(not identically zero )thatisanalyticin 1z andhaszerosatthe

followingindicatedsetofpoints?Whyorwhynot?

(i) 11

{ : }S n is a natural numbern

(ii) 21

{1 : }S n is a natural numbern

(iii) 3 { : 1, Re( ) 0}S z z z 41 1 1

( ) { : }2 2 2

iv S z iy y .